There are only two examples of nonlinear flows for which the
Floquet multipliers can be evaluated analytically.
Both are cheats.
One example is the 2-dimensional flow
\[
\begin{aligned}
\dot{q} &= ~p + q(1-q^2-p^2) \, , \\
\dot{p} &= -q + p(1-q^2-p^2) \, .
\end{aligned}
\]
It is easy to see that this flow has an equilibrium at the origin
\((p,\,q) = (0,\,0)\). Is this equilibrium stable or unstable?

Stable

Unstable

Q1.3 \(\quad\) (continuing from Question 2)

Go to polar coordinates
\( (q,p) = (r \cos \theta,r \sin \theta) \) and find
the limit cycle of this flow. What is the contracting Floquet exponent for this limit cycle?

-2

-1

0

1

2

Q1.4 \(\quad\) Stability of an equilibrium

In this question, you are going to compute stability eigenvalues and
eigenvectors of one of the equilibria of the Rössler system.

Start with completing definition of \( \mbox{StabilityMatrix(ssp)} \) in
\( \mbox{Rossler.py} \). Replace the 'None' elements of the matrix by the
appropriate partial derivatives.

The next task is to call this function from Stability.py.
See that in the beginning of Stability.py, we import
the Rossler module, where we now have the definition of the stability
matrix. Find the line where you are asked to evaluate this matrix at the \( \mbox{eq0} \), and
replace its `None' value by the stability matrix at \( \mbox{eq0} \). Functions of
other modules in python are called as follows
\[ \mbox{ModuleName.FunctionName(arguments)} \]
Read through the rest of the code and complete the lines where you see the
\( \mbox{#COMPLETE THIS LINE}\)
comment. Once you finish, run \( \mbox{Stability.py} \) to see its output.

You should see a flow spiraling out on a plane that is spanned by the real
and imaginary parts of the expanding eigenvector. Our goal for showing this
to you is to illustrate the "locally linear" behavior of the flow. Here, we
pick an initial condition very close to the equilibrium, with a little
perturbation in the unstable plane.

When run, Stability.py should also print the stability
eigenvalues at the equilibrium in the terminal, type in the real
part \( Re \lambda_1 \)of the most expanding eigenvalue in the box below.
Please use at least 4 decimal digits in your answer.

Q1.5 \(\quad\) Stability of a periodic orbit

In this problem you are going to calculate the Floquet matrix
(the Jacobian for one period) and its eigenvalues and eigenvectors
for the shortest periodic orbit of the Rössler system.
In CycleStability.py, we have given you the initial condition
and period of this orbit. Remember that
the Jacobian satisfies the following differential equation
\[
\frac{d}{dt} J^t(x_0) = A(x) J(x_0)\, , x = f^t(x_0)\, , \quad
\mbox{initial condition}\, J^t(x_0) = \mathbf{1}\,,
\]
where \(A(x)\) is the stability matrix.
Note that both sides of the above differential equation are
\(d \times d\) matrices, and the value of \(A(x)\) depends on
where it is evaluated in the state space. In other words, we need to
evaluate the above equation along with the orbit. In order to be
able to integrate it using our generic integrators, we need to convert
this problem into a \(d + d\times d\) dimensional linear ODE,
where first \(d\) elements
are the state space points, and the remaining \( d \times d \) are the
elements of the Jacobian matrix. We have written the velocity
function for
this extended system in Rossler.py.

Begin this exercise by reading the content of
\( \mbox{JacobianVelocity(sspJacobian, t)} \) in \( \mbox{Rossler.py} \) and understand
its construction. You are going to integrate this function in
CycleStability.py.

Now go to CycleStability.py and complete the line where
you specify the initial condition for the Jacobian. Read through
the code and complete
the line where you need to find the eigenvalues and eigenvectors of the
Jacobian.

Run CycleStability.py. You should see the periodic orbit and
the Floquet vectors associated with it.
Which arrow corresponds to the marginal
direction?

Pink

Black

Red

Green

Yellow

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